This research demonstrates bounds on independence ratios in certain planar graphs, highlighting their properties.
We prove that every ‐vertex planar graph with no triangle sharing an edge with a 4‐cycle has independence ratio for . This result implies that the same bound holds for 4‐cycle‐free planar graphs and planar graphs with no adjacent triangles and no triangle sharing an edge with a 5‐cycle. For the latter case we strengthen the bound to .
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Kelly et al. (2026) studied this question.
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